From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4998 Path: news.gmane.org!not-for-mail From: Andrew Salch Newsgroups: gmane.science.mathematics.categories Subject: Re: Topology on cohomology groups Date: Fri, 19 Jun 2009 17:39:57 -0400 (EDT) Message-ID: Reply-To: Andrew Salch NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1245608690 9140 80.91.229.12 (21 Jun 2009 18:24:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 21 Jun 2009 18:24:50 +0000 (UTC) To: Steve Vickers , Original-X-From: categories@mta.ca Sun Jun 21 20:24:48 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MIRih-00078l-LC for gsmc-categories@m.gmane.org; Sun, 21 Jun 2009 20:24:47 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MIR0y-0006QR-9j for categories-list@mta.ca; Sun, 21 Jun 2009 14:39:36 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4998 Archived-At: This certain does happen--if E^* is a generalized cohomology theory, and X is an infinite complex, we often want to topologize E^*(X) by the inverse limit of the E-cohomology of the finite skeleta of X. This is why, for example, if E is a complex-oriented cohomology theory, the E-cohomology of infinite-dimensional complex projective space is a power series ring (and not merely a polynomial ring), in one variable, over the coefficient ring E^*. This is sometimes important and useful in topology (for example, in the situation above, where one uses the above description of E^*(CP^{\infty}) to associate a 1-dimensional formal group law to E), and sometimes it's more just a hassle: for example, the early papers on the Adams-Novikov spectral sequence used MU-cohomology (i.e., complex cobordism), and this necessarily meant keeping track of the topology on MU^*(X) of various spectra E, since for example MU^*(MU), the ring of stable natural transformations of MU^*, has infinite homogeneous sums, and one had to handle completed tensor products of MU^*(MU)-modules; the modern way is to use generalized homology instead of generalized cohomology for these generalized Adams spectral sequences, which does away with the topologies and the need for completed tensor products (of course, the price one pays is that one is then, in the case of MU, dealing with MU_*(MU)-comodules rather than (topological) MU^*(MU)-modules, and computing Cotor rather than Ext; but this seems to be worth it). There's some discussion of this in Ravenel's green book. The paper on unstable operations by Boardman, Johnson, and Wilson in the Handbook of Algebraic Topology also includes some discussion and some nice manipulations of topologies (again, coming from the finite skeleta of an infinite complex) on some generalized cohomology rings and modules. There are also generalized homology theories, like the Morava E-theories, which occur as completions of other generalized homology theories, and so E_*(X) naturally has a topology (coming from the completion) when E is one of these theories; recent developments in stable homotopy theory make it seem likely that there will be more such theories in our future. Hope this is useful to you, Andrew S. On Fri, 19 Jun 2009, Steve Vickers wrote: > A cohomology group can easily be an infinite power of the coefficient > group. But such a group has a natural non-discrete topology, namely the > compact-open (which in this case is also the product topology). > > Are there approaches to cohomology that, as part of the process, also > supply topologies on the cohomology groups? > > [I'm trying to understand the topos-theoretic account of cohomology as > in Johnstone's "Topos Theory". But it looks heavily dependent on having > a classical base topos, since it uses the classical proof of sufficiency > of injectives (together with the existence of Barr covers) to deduce the > same property internally in any Grothendieck topos. For a more fully > constructive theory I wonder if one needs to take better care of the > topologies.] > > Steve Vickers. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]